Algebra
Axis of Symmetry – The Mirror Line Through a Parabola
The axis of symmetry is the vertical line passing through a parabola's vertex, splitting the curve into two perfectly mirrored halves. It shares the exact same formula as the vertex's x-coordinate, x = −b ÷ (2a) – the difference is really just in how you use the number: the vertex is a single point, while the axis of symmetry is the entire vertical line through it.
Because of this mirror symmetry, any point on a parabola has a matching point the same distance from the axis on the opposite side, at exactly the same height. This is a powerful shortcut: once you know one point on a parabola, you instantly know a second one for free.
Using the Axis of Symmetry
The axis of symmetry is x = −b ÷ (2a). Points equidistant from the axis, on opposite sides, share the same y-value.
On y = x² − 4x + 3, the axis of symmetry is the vertical mirror line x = 2. Every point on the curve has a matching “reflection” point the same distance away on the other side, at the exact same height.
The red pair — (1, 0) and (3, 0) — are both 1 unit from the axis and share the same height (y = 0). The orange pair — (0, 3) and (4, 3) — are both 2 units from the axis and share the same height (y = 3). This mirroring is exactly what “axis of symmetry” means: fold the graph along that vertical line, and the two halves of the curve land perfectly on top of each other.
x = −(−6) ÷ (2 × 1) = 6 ÷ 2 = x = 3.
The point x = 1 is 3 units left of the axis, so the mirror point is 3 units right: 4 + 3 = 7.
Real-Life Application
- Bridge design: the axis of symmetry helps engineers balance a parabolic arch evenly.
- Sports: a ball's flight path is symmetric about its highest point.
- Architecture: parabolic archways are often designed to be symmetric about a central line.
Key Takeaways
- The axis of symmetry is the vertical line x = −b ÷ (2a) through the vertex.
- It splits the parabola into two mirror-image halves.
- Points equidistant from the axis share the same y-value.